Random Topology Power Grid Modeling and Automated ...Review,&Aug&506,&2014&& 1/31 Random Topology Power Grid Modeling and Automated Simulation Platform Zhifang Wang Virginia Commonwealth

1/31 CERTS Review, Aug 5-‐6, 2014

Random Topology Power Grid Modeling and Automated

Simulation Platform

Zhifang Wang Virginia Commonwealth University

Robert J. Thomas Cornell University


Project Objectives

u Complete a comprehensive study of realistic power grid topology and parameters.

u Formulate appropriate random topology power grid models.

u Provide an automated simulation platform  enable sufficient testing and verification of

new concepts, methods, and controls in power grids

 compatible with MATPOWER


Motivation of the Project

u Future power engineers and researchers need appropriate randomly generated grid network topologies to do genuine Monte Carlo experiments. Ø  If the random networks are truly representative and if the

concepts or methods test well in this environment they would test well on any instance of such a network.

u Current situation

 extremely difficult to obtain realistic system data from utilities

  limited reference test cases

 existing models with shortcomings, i.e. connectivity and scaling property


Critical Applications for the Grid

•  Renewable generation interconnection

•  PMU placements to facilitate fast state estimation and real-time state awareness

•  Transmission expansion planning

•  Grid vulnerability and security analysis •  Transient stability controls

•  Power market strategy experiments •  Smart grid communication infrastructure –  especially when the communication network

layout partially or fully overlaps with the grid itself, e.g. power line communication.


Precursors of our study

•  Most of the literature has used one specific real grid or reference models for testing –  IEEE 30 57, 118 and 300 bus systems –  Power systems test case archive: http://www.ee.washington.edu/research/pstca/

•  Scalable models to grasp macroscopic trends –  [Parashar and Thorp ’04] ring topology –  [Rosas-Casals, Valverde, Solé ’07] tree topology

–  [Watts, Strogatz ’98], small-world topology

– Other models


Electric Power Grid Network

•  3 sections in transmission – High, Medium and Low

voltage sections

Transmission

Distribu.on


Admittance matrix and the graph topology

•  Line-Node Incidence Matrix A (M x N):

Line m: node i – node j →

•  Admittance matrix

•  Graph Laplacian:

•  Observation: Y is a complex-weighted Laplacian

•  Complex weights given by the admittances of the lines

Y = AT diag(y1, . . . , yM )A

L = ATA

yl = 1/zl = 1/(rl + jxl)


The Laws for the Grid

•  Voltage, Currents, Powers à narrow spectrum

•  “phasors” (complex numbers whose phase and amplitude match the AC signal V or I), e.g.

•  Kirchhoff’s Voltage/Current laws (KVL-KCL)

•  Ohm’s law

Alterna;ng Current(AC) ~frequency 60 or 50 Hz

X

i2Circuit

Vi = 0,X

i2Node

Ii = 0

Vi = ZiIi

f0

V = |V | exp(j✓)v(t) =p(2)|V | exp(j✓) exp(j2⇡f0t)


Statistical Modeling of Power Grid

•  Topological and electrical characteristics of the transmission grid

•  As captured by the statistical properties of the grid admittance matrix Y

•  Propose a random topology power grid model with plausible topology and electrical parameters

Electricity

Gen

erators

Loads

Power Grid


Statistical Modeling of Power Grid

Topology Electrical

Parameters

Rand-topo Power Grid Model


Power Grid – Network Topology

Topology

•  Small-world Properties •  Node Degree Distribution •  Connectivity Scaling •  etc


Power Grid – Electrical Parameters

•  The line impedance distribution – heavy-tailed

Electrical Parameters


Random topology models

•  ‘98 Watts and Strogatz, Nature

– Conjecture: Power Grids are small world networks

– Topological studies: [Newman ’03], [Whitney & Alderson’06],[Wang, Rong,’09]

– Degree distribution: [Albert et al. ‘04],[Rosas-Casals et al. ‘07]

Erdos Reny Small World Power network


Small world : high clustering coefficient

•  high average clustering coefficient of the sample power grid network examined

Defini;on of clustering coefficient

#vi

vj vkCi =2|{ejk}|ki(ki � 1)

8 (vj , vk) 2 Ni, ejk 2 E

(Ni = neighbors of vi)

(E = edges)

(ki = degree of i)

Random graph ( Erdös Rényi)

Grid


Small world: short average path length

•  Observation:

•  Not bad to overlay communications with the lines – relatively short distance

N: Number of nodes m: number of lines <k> Average Degree <l> Average shortest

path length ρ Pearson Coefficient R{ } Ra;o of nodes

with larger nodal degree

< l >⇡ 3 log(N)

k k>


Degree distribution

•  Most researchers assume that power grid node degree has a Geometric PDF

– e.g.[Albert et al. ’04,Rosas-Casals’07]

•  Way to highlight: Probability Generating Function (PGF)

–  For a mixture model

Our analysis result 1.The degree distribu;on is a mixture of a truncated exponen;al and finite support random variable

2.The average degree vs. N is O(1)

GX(z) = E(zX) =X

k

p(X=k)zk


Results (a)  All buses (b)  Gen buses (c)  Load buses. (d)  Connec.on buses. (e) Gen+Load buses. The zeros are red ’+’

Degree of Generator buses

Degree of Load buses

Degree of Connec;on buses PGF NYISO

data


Small World conjecture

•  Some evidence contradicting it

– For a SW network with N nodes, to guarantee with high probability a connected network (no isolated component) the scaling laws for the average degree <k> >> log N

– The average degree in power grids is ~ constant (3-4)


Scaling Algebraic Connectivity

•  Graph Laplacian second smallest eigenvalue

•  Values shown in 2 D regular graph k=4 and k=3

1 D regular graph k=4 and k=2


Plausible topology

•  The proposed model that matches this trend is what we call nested-Small-world graph

–  IEEE à SW subnet 30; NYISO & WECC à SW sub-net 300

IEEE 300: Correlated rewiring

SW: independent rewiring


Impedance distribution

•  Absolute values of the impedances

•  Prevailingly heavy tailed distributions

•  NYISO best fit à clipped Double Pareto Log-normal

Zpr = R + jX ⇡ jX


Impedance attribution

•  Impedance grows with distance: local à short; rewires à medium; lattice connections à long lines

•  RT-nestedSmallWorld: developed a Matlab code pkg – a helpful tool for many other researchers in the area.


Bus Type Assignment

•  Three bus types in a grid:

• Generation bus •  Load bus

• Connection bus •  RT-nestedSmallWorld: random assignment

according to the bus type ratios


Bus Type vs. Node degree

•  Correlation between bus types and node degree


Bus Type vs. Clustering Coefficient


The Stochastic Model for Cascading Process

Model the line states as conditionally Markovian given the line flows ``crossing’’ times

B’(t) incoporates the grid network state

Line State transi;on: line tripping vs. restora;on

The crossing intervals of line flow F(t) determine the transi;on rates: λ0(t)-‐random con;ngency,

λ*(t)-‐overload line tripping rate

ܤᇱ()ܤ

ଵᇱ(ଵ)ܤ

ܤᇱ(ଵ)

ᇱ(ଵ)ܤ

,ଵܤᇱ(ଶ)

,ିଵܤᇱ(ଶ)

,ܤᇱ(ଶ)

ଵ()ߣ

ߣ()

()ߣ

ିଵߣ(ଵ)

ଵ(ଵ)ߣ

(ଵ)ߣStage 0 (intact)

Stage 1 (1 line lost)

Stage 2 (2 lines lost)

Line ac;ve Line tripped

the overload threshold


Flow model and level crossing statistics

Gaussian model for the flows (tractability)

Gaussian sta;s;cs of genera;on and loads We assume loads are independent spa;ally, non-‐sta;onary (cyclosta;onary) Temporal correla;on of the load is properly captured

The sta;s;cs of line flows

Gaussian process à Rice’s result (1958) Sta;s;cs of level-‐crossing intervals èline state transi;on rate èaverage life;me èprobability of overload and distances metrics

The power grid DC flow model

At – the line-‐node incidence matrix yl – the line admimance


–  Expected number of flow crossings after which the line gets tripped

– 

–  If it is beyond the coherence time of the process, the non stationary metric can be computed using the worst case parameters

Expected Safety Time of a Line


Bus Type Assignment vs. Grid Vulnerability

IEEE-300 bus system,

given the same topology and generation and load statistical settings, the test cases with random bus type assignments tend to have larger expected safety time than that of the realistic grid settings.


Conclusions

•  The admittance matrix of power grids has peculiar features that follow clear statistical trends.

•  RT-nestedSmallWorld model is designed to capture the accurate statistical properties of real grid topology and electrical parameters.

•  It is critical to understand how to design the RT-power grid model better.

•  Random assignment of bus types will be replaced with a more accurate one which is consistent with that of realistic grids.


Questions? J

Thank You!

Documents

Random Topology Power Grid Modeling and Automated ...Review,&Aug&506,&2014&& 1/31 Random Topology Power Grid Modeling and Automated Simulation Platform Zhifang Wang Virginia Commonwealth